Recall that an open interval (a, b) on the real line R is defined as (a,b){re...

Transcribed Image Text:

Recall that an open interval (a, b) on the real line R is defined as (a,b){re Ra< x <b}. For a fixed n N = {1,2,3,4,...}, we define the following open interval: = (^— ²¹, 1). n an infinite family of open intervals (Sn)neN. Sn = Letting n range over N, we obtain (i) Prove that for all n E N, S, Sn+1. (ii) Prove that UnEN Sn = (0,1), where Explain your answer. US₁ S₁ US₂ US3U... = NEN (iii) What is EN Snl (i.e. the cardinal of EN Sn) where Sn=S₁nS₂n San...? NEN Recall that an open interval (a, b) on the real line R is defined as (a,b){re Ra< x <b}. For a fixed n N = {1,2,3,4,...}, we define the following open interval: = (^— ²¹, 1). n an infinite family of open intervals (Sn)neN. Sn = Letting n range over N, we obtain (i) Prove that for all n E N, S, Sn+1. (ii) Prove that UnEN Sn = (0,1), where Explain your answer. US₁ S₁ US₂ US3U... = NEN (iii) What is EN Snl (i.e. the cardinal of EN Sn) where Sn=S₁nS₂n San...? NEN